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Theorem funbrfv 5495
Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
funbrfv  |-  ( Fun 
F  ->  ( A F B  ->  ( F `
 A )  =  B ) )

Proof of Theorem funbrfv
StepHypRef Expression
1 funrel 5211 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 brrelex2 4716 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  B  e.  _V )
31, 2sylan 459 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  B  e.  _V )
4 breq2 4001 . . . . . 6  |-  ( y  =  B  ->  ( A F y  <->  A F B ) )
54anbi2d 687 . . . . 5  |-  ( y  =  B  ->  (
( Fun  F  /\  A F y )  <->  ( Fun  F  /\  A F B ) ) )
6 eqeq2 2267 . . . . 5  |-  ( y  =  B  ->  (
( F `  A
)  =  y  <->  ( F `  A )  =  B ) )
75, 6imbi12d 313 . . . 4  |-  ( y  =  B  ->  (
( ( Fun  F  /\  A F y )  ->  ( F `  A )  =  y )  <->  ( ( Fun 
F  /\  A F B )  ->  ( F `  A )  =  B ) ) )
8 funeu 5217 . . . . . 6  |-  ( ( Fun  F  /\  A F y )  ->  E! y  A F
y )
9 tz6.12-1 5477 . . . . . 6  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
108, 9sylan2 462 . . . . 5  |-  ( ( A F y  /\  ( Fun  F  /\  A F y ) )  ->  ( F `  A )  =  y )
1110anabss7 797 . . . 4  |-  ( ( Fun  F  /\  A F y )  -> 
( F `  A
)  =  y )
127, 11vtoclg 2818 . . 3  |-  ( B  e.  _V  ->  (
( Fun  F  /\  A F B )  -> 
( F `  A
)  =  B ) )
133, 12mpcom 34 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( F `  A )  =  B )
1413ex 425 1  |-  ( Fun 
F  ->  ( A F B  ->  ( F `
 A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   E!weu 2118   _Vcvv 2763   class class class wbr 3997   Rel wrel 4666   Fun wfun 4667   ` cfv 4673
This theorem is referenced by:  funopfv  5496  fnbrfvb  5497  fvelima  5508  fvi  5513  fmptco  5625  fliftfun  5745  fliftval  5749  opabiota  6259  fpwwe2  8233  nqerid  8525  sum0  12160  sumz  12161  fsumsers  12167  isumclim  12186  dvadd  19252  dvmul  19253  dvco  19259  dvcj  19262  dvrec  19267  dvcnv  19287  dvef  19290  ftc1cn  19353  ulmdv  19743  minvecolem4b  21418  minvecolem4  21420  hlimuni  21779  chscllem4  22180  fvtransport  24031  fvray  24140  fvline  24143
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fv 4689
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